Question

This system consists of a marble rolling down a ramp. Assume the mass of the marble is 5.0 g. Decide what would be needed to measure or determine the other quantities to calculate gravitational potential energy and kinetic energy. Solve the equation for the speed of the marble at the bottom of the ramp and compare it to the experimentally measured value.

Answer 1

Answer: v = √2.g.h

Explanation:

If we assume that the marble can be approximated by a point mass, and that it starts from rest at a height h, at that moment, all the energy of the system will be gravitational potential energy, that can be written as follows:

U₁ = m. g. h

As we know m, and g is a constant equal to 9.8 m/s², we will need to measure  the height h, either directly, or in an indirect way from the value of the angle that the ramp does with the horizontal, and the measured value of  the distance travelled along the ramp, x.

So, we could write U₁ as follows:

U₁ = m . g. x. sin θ

Now, at the bottom of the ramp, neglecting fricition, all this potential energy must become kinetic energy, as follows:

U₁ = K₂ ⇒ mgh = 1/2 m(v₂)²

Simplifying and solving for v₂ (the speed of the marble at the top of the bottom), we have:

v₂ = √2.g.h

Once the marble has reached to the bottom of the ramp, it has no more net  external forces acting on it (neglecting friction), it must continue moving at constant speed, equal to v₂.

This value can be measured easily, measuring the displacement between 2 points, and the time used to pass between those points, and computing v₂ as follows:

v₂ = Δx / Δt

If the measured value is different to the one calcultated (beyond the expected experimental error) this means that the friction was not so negligible.